Editing General equilibrium theory (section)

====Nonconvexities in large economies====
{{Main|Shapley–Folkman lemma}}
Starr (1969) applied the [[Shapley–Folkman lemma|Shapley–Folkman–Starr theorem]]<!-- his corollary to the [[Shapley–Folkman lemma|Shapley–Folkman theorem]] --> to prove that even without [[convex preferences]] there exists an approximate equilibrium. The Shapley–Folkman–Starr results bound the distance from an "approximate" [[economic equilibrium]] to an equilibrium of a "convexified" economy, when the number of agents exceeds the dimension of the goods.<ref name="s69">{{Cite journal |doi=10.2307/1909201 |last=Starr |first=Ross M. |author-link=Ross Starr |issue=1 |journal=Econometrica |pages=25–38 |title=Quasi-equilibria in markets with non-convex preferences |jstor=1909201 |volume=37 |year=1969 |url=http://econ.ucsd.edu/~rstarr/Non-Convex%20Preferences.pdf |archive-url=https://web.archive.org/web/20170809014202/http://econ.ucsd.edu/~rstarr/Non-Convex%20Preferences.pdf |archive-date=2017-08-09 |url-status=live |citeseerx=10.1.1.297.8498 }}</ref> Following Starr's paper, the Shapley–Folkman–Starr results were "much exploited in the theoretical literature", according to Guesnerie,<ref name="Guesnerie1989">{{cite book |last=Guesnerie |first=Roger |year=1989 |chapter=First-best allocation of resources with nonconvexities in production|pages=99–143|editor=Bernard Cornet and Henry Tulkens |title= Contributions to Operations Research and Economics: The twentieth anniversary of CORE (Papers from the symposium held in Louvain-la-Neuve, January 1987) |publisher=MIT Press |location=Cambridge, MA |isbn=978-0-262-03149-3 |mr=1104662}}</ref>{{rp|112}} who wrote the following:
<blockquote>
some key results obtained under the convexity assumption remain (approximately) relevant in circumstances where convexity fails. For example, in economies with a large consumption side, nonconvexities in preferences do not destroy the standard results of, say Debreu's theory of value. In the same way, if indivisibilities in the production sector are small with respect to the size of the economy, [ . . . ] then standard results are affected in only a minor way.<ref name="Guesnerie1989"/>{{rp|99}}
</blockquote>
To this text, Guesnerie appended the following footnote:
<blockquote>
The derivation of these results in general form has been one of the major achievements of postwar economic theory.<ref name="Guesnerie1989" />{{rp|138}}
</blockquote>
In particular, the Shapley-Folkman-Starr results were incorporated in the theory of general economic equilibria<ref>See pages 392–399 for the Shapley-Folkman-Starr results and see p. 188 for applications in {{cite book|last1=Arrow|first1=Kenneth J.|author-link1=Kenneth J. Arrow|last2=Hahn|first2=Frank H.|author-link2=Frank Hahn|year=1971|chapter=Appendix B: Convex and related sets|title=General Competitive Analysis|url=https://archive.org/details/generalcompetiti0000arro|url-access=registration|publisher=Holden-Day [North-Holland]|pages=[https://archive.org/details/generalcompetiti0000arro/page/375 375–401]|mr=439057|series=Mathematical economics texts [Advanced textbooks in economics]|location=San Francisco|isbn=978-0-444-85497-1|issue=6 [12]}}</ref><ref>Pages 52–55 with applications on pages 145–146, 152–153, and 274–275 in {{cite book|last=Mas-Colell|first=Andreu|author-link=Andreu Mas-Colell|year=1985|chapter=1.L Averages of sets| title=The Theory of General Economic Equilibrium: A ''Differentiable'' Approach
| series=Econometric Society Monographs|publisher=Cambridge University Press|isbn=978-0-521-26514-0
|mr=1113262|issue=9}}</ref><ref>{{cite book|last=Hildenbrand|first=Werner |title=Core and Equilibria of a Large Economy|series=Princeton Studies in Mathematical Economics|publisher=Princeton University Press|location=Princeton, New Jersey|year=1974 |pages=viii+251|isbn=978-0-691-04189-6|mr=389160|issue=5}}</ref> and in the [[microeconomics|theory]] of [[market failure]]s<ref>See section 7.2 Convexification by numbers in Salanié: {{cite book|last=Salanié|first=Bernard |chapter=7 Nonconvexities|title=Microeconomics of market failures|edition=English translation of the (1998) French ''Microéconomie: Les défaillances du marché'' (Economica, Paris)|year=2000|publisher=MIT Press|location=Cambridge, Massachusetts|pages=107–125|isbn=978-0-262-19443-3}}</ref> and of [[public economics]].<ref>An "informal" presentation appears in pages 63–65 of Laffont: {{cite book|last=Laffont|first=Jean-Jacques|author-link=Jean-Jacques Laffont|year=1988|chapter=3 Nonconvexities|title=Fundamentals of Public Economics|url=https://archive.org/details/fundamentalsofpu0000laff|url-access=registration|publisher=MIT|isbn=978-0-585-13445-1}}</ref>